My guesses:
1. Longest game ever played had 350 moves.
2. Shortest serious game ever played had 9 moves.
3. Longest series of moves by one piece is 35 moves.
4. The last capture happened on the 290th move.
5. Longest finished games without captures was 80 moves.
Am I close? 
Close enough, but lets w8 for few more answers. 
#1The longest is 268 or 269 by Nicolic and someone else.
#2The shortest one is around 10 moves
#3 no idea
#4 somewhere after the 90th, if I recall correctly (if you're talking about the latest first capture)
#5 is the Nuber-Keckeisen from the mid-ninties lasting 34 moves
1. 540
2. 1 move.
3. Anand's catalan game in W Cup 2010 had 10 moves from the knight.
Hmm.. Looking into it, I actually found a game with a fool's mate played by L. Darling and R. Wood in 1983. Mr. Wood was an USCF master who apparently died young and who only has 2 games on display at chessgames.com totalling his efforts to 19 moves for 2 victories.
Apparently mr. Darling was participating in the mind sports olympiad as some sort of guru on the card game Magic: The Gathering.
1. 350 moves.
2. 8 moves.
3. 10 moves by Annad in the WCC with his knight
4. 285th move.
5. 75 moves.
| 269 | I. Nikolic - Arsovic | ½-½ | Belgrade 1989 | |
| 237 | Fressinet - Kosteniuk | 0-1 | Villandry 2007, rapid | |
| 227 | T. L. Petrosjan - Sammour Hasbun | ½-½ | Dos Hermanas Internet 2007, blitz | |
| 215 | Ivanchuk - Leko | ½-½ | World Blitz Ch., Moscow 2007 | |
| 209 | Chekhlov - Stavrinov * | ½-½ | Riga, City Ch. 1988 | |
| 202 | Gil-galad - izit | 0-1 | ICC team 45 45 | |
| 200 | Wegner - Johnsen | ½-½ | Gausdal 1991 | |
| 200 | Meiers - Rausis * | ½-½ | Riga, Ch. Latvia 1989 | |
| 199 | T. L. Petrosjan - Milanovic | 0-1 | Dos Hermanas Internet 2005, blitz | |
| 196 | Murey - Rausis | ½-½ | Paris 1995 |
This position occurred more often, and White did not always resign; in Mory - Hareux, Besançon 1999, he even drew.
3. Longest series of moves by one piece: 80
Black to play
Yueh Wei Po - Arora
Olympiad Turin, 2006
67...Rb4+ was the first of 80 consecutive Rook moves after the last of which, at move 147, a draw was agreed.
This record had long been in the hands of a Queen from the times of Queen Victoria, with 73 consecutive moves. 
Black to play
Mackenzie - Mason
London 1882
From 72...Qd1 to 144...Qf4, when a draw was agreed, the black Queen never had a moment's rest.
Latest first capture: 94
White to play
Rogoff - Williams
Stockholm, World Junior ch, 1969
94.bxc5 finally broke the spell. The game is reputed to have ended in a draw at move 221, but no moves beyond move 106 have surfaced.
5. What is longest finished games without captures?
In two cases, the game ended before there was any capture.
White to play
Meijfroidt - Lenoir
Veurne 2000
White overstepped the time limit here at move 72.
White to play
Filipowicz - Smederevac
Polanica Zdroj, 1966
In this position after Black's 70th move, the game was declared a draw.
According to Filipowicz, he had offered a draw twice in vain, and now he baffled his opponent byclaiming a draw on the ground of the 50-move rule - the most crowded position in which that rule was ever invoked.
There are full games for most of this facts, but not pgn, so I would have to insert move by move. But if you really really want to see some of this games in full, I will do it.
Next.
Chapter 1: moves
6. What is longest decisive game without captures?
7. What was quickest exchange game? (after how many moves all pieces were captured - not including pawns)
8. What is longest series of checks in one game? (check after check after check)
9. What is greatest number of checks in one game by both sides?
10. What was earliest double polygamy? (after how many moves did both sides have 2 queens on the board)
11. What is greatest number of castlings in one game?
1. 192 moves
2. 9 move resignation (lost by tournament leader on that round)
3. No idea (few perpetual checks by queen lasted to 10-15 moves and draw agreed)
4. No idea
5. No idea
6. Fools mate had happened for sure.
7. No idea
8. 4 consective check have happened. Couldn't recollect the game
9. No idea
10. In one of the game after 54 moves both side had 2 queens
11. What?
There are full games for most of this facts, but not pgn, so I would have to insert move by move. But if you really really want to see some of this games in full, I will do it.
Next.
Chapter 1: moves
6. What is longest decisive game without captures?
7. What was quickest exchange game? (after how many moves all pieces were captured - not including pawns)
8. What is longest series of checks in one game? (check after check after check)
9. What is greatest number of checks in one game by both sides?
10. What was earliest double polygamy? (after how many moves did both sides have 2 queens on the board)
11. What is greatest number of castlings in one game?
10. If live chess is included, My game had 2 queens each on move 13.
for 3: I saw (and posted somewhere in these forums) a game with 3 Castlings(? is that a word?), if I remember correctly white castled K side, Black Castled K side and many moves later White castled Q side.
for 3: I saw (and posted somewhere in these forums) a game with 3 Castlings(? is that a word?), if I remember correctly white castled K side, Black Castled K side and many moves later White castled Q side.
Thats illegal. This question does not make sense to me.
What the other player was doing at the third castling? sleeping?
A mathematician I knew once wrote an article working out the problem of what would happen if there were four "doubled" White pawns lined up in the middle of the e-file. I couldn't keep my eyes open when he explained the solution to me. It got published some place....
Ok, this was not about your games, but more like records form GM games in history. And yes, there was game with 3 castlings, where both players, and officials missed to see that.
Greatest number of castlings: 3
Heidenfeld - Kerins, Dublin 1973
1.e4 e6 2.d4 d5 3.Be3 Nf6 4.e5 Nfd7 5.f4 c5 6.c3 Nc6 7.Nf3 Qb6 8.Qd2 c4 9.Be2 Na5 10.O-O f5 11.Ng5 Be7 12.g4 Bxg5 13.fxg5 Nf8 14.gxf5 exf5 15.Bf3 Be6 16.Qg2 O-O-O 17.Na3 Ng6 18.Qd2 f4 19.Bf2 Bh3 20.Rfb1 Bf5 21.Nc2 h6 22.gxh6 Rxh6 23.Nb4 Qe6 24.Qe2 Ne7 25.b3 Qg6+ 26.Kf1 Bxb1 27.bxc4 dxc4 28.Qb2 Bd3+ 29.Ke1 Be4 30.Qe2 Bxf3 31.Qxf3 Rxh2 32.d5 Qf5 33.O-O-O Rh3 34.Qe2 Rxc3+ 35.Kb2 Rh3 36.d6 Nec6 37.Nxc6 Nxc6 38.e6 Qe5+ 39.Qxe5 Nxe5 40.d7+ Nxd7 0-1
 Position after 9...Na5 |
 Position after 16.Qg2 |
 Position after 32...Qf5 |
What's wrong with four pawns on the same file? You can even get six pawns on the same file...I should know, I was participating in a thread working out the fastest way to achieve that a couple of months back.
But it's a pity, I guess I'll have to refrain from this contest...I know where you're getting your info from, Streptomicin. (The board style is distinctive.) :P
I found some interesting chess facts. I was surprised to see the answer to each of this questions. I wont ask them all at once, and do not try to google the answers, there is no prize, I just want to know what do you think, as I was at least 40 moves away from correct answer every time.
chapter 1: moves